Extended States in a Lifshitz Tail Regime for Random Schr\"odinger Operators on Trees

TL;DR

This paper investigates the presence of extended states in a Bethe lattice with random potential, revealing that disorder-induced resonances lead to extended states beyond the typical energy band, especially in low-density Lifshits tail regimes.

Contribution

It demonstrates the existence of extended states caused by disorder resonances outside the main energy band on Bethe lattices, addressing a key question in Anderson localization.

Findings

Extended states occur beyond the hopping energy band.

Disorder enables resonances leading to extended states.

Extended states are present in Lifshits tail regimes at low density.

Abstract

We resolve an existing question concerning the location of the mobility edge for operators with a hopping term and a random potential on the Bethe lattice. The model has been among the earliest studied for Anderson localization, and it continues to attract attention because of analogies which have been suggested with localization issues for many particle systems. We find that extended states appear through disorder enabled resonances well beyond the energy band of the operator's hopping term. For weak disorder this includes a Lifshits tail regime of very low density of states.